REPRESENTATION AND REALIZATION OF BOUNDED HOLOMORPHIC FUNCTIONS DEFINED ON A POLYDOMAIN By ANDREW T. TOMERLIN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2000 Dedicated to my parents Dr. Arthur H. Tomerlin and Mireya M. Tomerlin ACKNOWLEDGEMENTS First, I would like to thank my advisor, Dr. Scott A. McCullough, for his guidance toward the preparation of this dissertation. His insight, support, and pa- tience are deeply appreciated. Working with such an excellent advisor was an honor for me. I would also like to thank all of my committee members: Dr. Jacob Hammer, Dr. Murali Rao, Dr. Li-Chien Shen, and Dr. Douglas Cenzer. Their insight and support are also appreciated. I would like to thank most my parents and Paromita. They have been my inspiration, support, and life. Without them, I would not have accomplished this work. II! PREFACE This work combines two sciences, mathematics and control theory. We assume the reader has basic knowledge in functional and complex analysis. Some knowledge of state variable methods would be helpful, but not necessary. Let us define some basic terminology used throughout the text. We define a region as an open connected subset of C" = {(21, . ..,«„) : *t £ C} where C is the complex plane. A kernel defined on a region ft is a function k : Vt x f} -> C. Define the unit disk D = {z : \z\ < 1} and the polydisk 0* = {{z = (z u ...,z d ) : z t g D}. IV TABLE OF CONTENTS ACKNOWLEDGEMENTS iii PREFACE iv ABSTRACT vii CHAPTERS 1 INTRODUCTION 1 2 REALIZATION AND REPRESENTATION 5 2.1 Representation and Realization on 0* 5 2.2 Transfer Function Embedding Theorem 7 3 REALIZATION AND REPRESENTATION ON A POLYDOMAIN . 10 3.1 Realization on Q, = fii x • • • x Q^ 10 3.2 Representation on fi = Oi x • • • x fi^ 13 4 INTERPOLATION 19 4.1 Interpolation on P* 19 4.2 Interpolation on fi = fix x • • • x f^ 20 5 CORONA THEOREMS 23 5.1 Toeplitz Corona Theorem on ID^ 23 5.2 Toeplitz Corona Theorem on fi = (]i x • • • x Hj 24 6 SYSTEMS THEORY 28 6.1 Roesser Model 28 6.2 Energy Conservative/ Energy Dissipative Linear Multidimensional Systems 30 6.3 Minimal Realizations 30 7 CONCLUSION 41 REFERENCES 43 BIOGRAPHICAL SKETCH 45 VI Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy REPRESENTATION AND REALIZATION OF BOUNDED HOLOMORPHIC FUNCTIONS DEFINED ON A POLYDOMAIN By Andrew T. Tomerlin December 2000 Chairman: Dr. Scott McCullough Major Department: Mathematics We define a Schur class of functions in terms of a kernel factorization property. We characterize the elements of this Schur class as transfer functions of a certain type of unitary colligation and as contractive functions over a class of operators acting on a Hilbert space. With these results we establish a Nevanlinna-Pick interpolation the- orem with our Schur class as the interpolating set. Further, we prove an extended version of the Toeplitz corona theorem. We examine our results within a systems theory framework and devise an efficient algorithm to realize a function of two vari- ables. vn CHAPTER 1 INTRODUCTION An analytic function / denned on the unit disc D = {z : \z\ < 1} is in H°°(p) provided ||/|| = {\f{z)\ : ; £ D} < oo. The study of bounded functions plays a decisive role in electrical engineering. Rational functions H(z) in H°°(B) are in one to one correspondence with linear time invariant stable physical systems. The rational function that is identified with a particular stable system is the transfer function of the system. Transfer functions are an input to output model of a physical system; thus knowledge of the properties of H°°(B), in particular the unit ball in H°°(B), is important for the study of linear time invariant stable systems. Denote the unit ball in ff»(D) as BH°°(B) = {/ € H°°{B) : ||/|| < 1}. Ele- ments of BH°°(B) are transfer functions of unitary colligations. A unitary colligation E is a tuple ([/,%,£,£*), where W, £,£* are Hilbert spaces and U = is a unitary operator. The transfer function associated to E is W^(z) = D + (I — zA)- l zB [2, 4]. A function / is in BH°°{B) if and only if / = VK E for some unitary colligation E. Let s(z,w) = t-^= denote the Szeeo kernel. The classical Nevanlinna-Pick Theorem implies that a complex valued function / defined on D is in BH°°(3) if and only if the kernel A'/iDxD^C, 'A B C D c_ -> H _C_ l-f(z)f(w) hj{z,w) = — — , (1.1) is positive semidefinite. Rewriting identity (1.1), it follows that / is in BH 00 ^) if and only if it has the factorization property: there exists a positive kernel K such that l-f(z)Ji^) = K(z,w)s(z,iv). (1.2) Fix a positive integer d, regions tlj, and positive kernels kj over Qj, j = 1, 2, . . . , d, and assume that, for each j, the reciprocal of kj has one positive square, i.e., 1 for some scalar valued functions b\, . . . ,&£ . Let fi denote the polyregion Qj x fta x • • • x (Id and note that the kernel k(z,w) = fei(zi, ^1)^2(22,^2) • ■ • kd(z d , W4) is a positive kernel over il. Given Hilbert spaces £,£*, £(£,£*) denotes the space of bounded linear operators from £ to £*. Define ^jfe(£,£») to be the set of £(£,£*) valued functions VF(z) = W(zi, 22, • • • , Zd) which are defined on fi and which satisfy the factorization property: there exist auxiliary Hilbert spaces Ci,...,Cd and functions Hi, . . . , Hd defined on f) with values in £(Cj, £*) such that d }=1 3 When kj — s for each j, Tk{£, £*) is known as the generalized Schur class [9] and is denoted Td{£,£*)- Elements of J~d{£, £*) have a factorization form TVd'- d TV d : /«. - W(z)W(wy = T -(z 3 ,w 3 )H 3 {z)H 3 {wy. . ' s It is evident from TV d that f,(C,C) = BH°°(B). In addition to the factorization property elements of J-d{£, £*) and $iy°°(D) share two other fundamental properties. First, functions in J~d{£, £*) are transfer functions of appropriately defined unitary colligations. Define a (/-variable unitary operator colligation S to be a tuple £ = (U,'H, £,£*), where %,£,£„ are Hilbert spaces, % = ® d j =x Hj has a fixed d-fold orthogonal decomposition, and U is a unitary operator u A B C D n The transfer function of the d- variable unitary operator colligation S is defined to be the operator-valued function defined on IF W z (z) = D + C(I- Z(z)A)- 1 Z(z)B where Z : B 1 -> U is defined by Z(z) = ®f =1 Z l {z t ) and Zi : D -> U x is defined by Zi(z) — zl-u,- A function / analytic in B* is in ^(S.S*) if and only if / = Wy. for some d- variable unitary colligation £ [2]. The second fundamental property ?&(€, £,) shares with BH°°(B) is that under proper interpretation, elements of ^d(£, £») are contractive functions. A function / analytic in B 1 is in Td{£,£*) if and only if /(7 1 !, T 2 , . . . , Td) has operator norm at most one for every d-tuple of commuting strict contractions T = {T%, . . . , Tj) acting on a Hilbert space H [2]. Let ttj be a region for j = 1, . . . , d and define f2 = fii x • • • x Q,j. For each region flj there exists an associated positive kernel kj defined on $7 such that the reciprocal kernel has one positive square. Then J-k{£,£*) can viewed as the generalization of Fd{£, £*) from 0* to the polyregion tt. Elements of Fk(£,£*) and jF rf (£,£ + ) share parallel properties. Elements of Tk{£-, £*) are transfer functions of an associated class of unitary colligations. We also show elements of J-k{£, £*) are contractive functions over an abstract class of operators acting on a Hilbert space. The organization of the thesis is as follows: in chapter two, we examine the results characterizing elements in J-d(£,£*) [2]. We extend these results and prove transfer functions of a certain type of contraction colligation are in .Fd(£, £*). We call this result the Transfer Function Embedding Theorem. Chapter three deals with the characterization of elements in J-k(£,£*). We prove elements of Tk{£-,£») are transfer functions of an associated class of unitary colligations. Moreover, we show that for specified kernels, elements of Tk(S,€») are contractive functions defined over a class of operators acting on a Hilbert space. Chapter four deals with Nevanlinna-Pick interpolation. We present the clas- sical Nevanlinna-Pick interpolation theorem and the extension of this theorem to several variables [1]. Using the results found in chapter three we prove an extended Nevanlinna-Pick interpolation theorem with ^(£,8*) as the interpolating set. Chapter five deals with the Corona problem. We begin the chapter with a review of the Carleson Corona Theorem [11] and the Toeplitz Corona Theorem [22, 17]. We present the Toeplitz Corona Theorem for polydisk the 0* [9]. Using the results found in chapter three we prove our own extended version of the Toeplitz Corona Theorem for the kernel k defined on the polyregion ft = Qi x • • • x ft<*. In chapter six, we discuss the Transfer Function Embedding Theorem in a systems theory framework. We show how results characterizing elements of !Fd{£, 8*) can be viewed as physical laws for multidimensional systems. Furthermore, we discuss the minimal realization problem for functions of two variables and give an efficient algorithm to realize a function of two variables. CHAPTER 2 REALIZATION AND REPRESENTATION We begin this chapter defining the Schur class F d (£,£*). This set is defined as having a factorization property TV a. J. Agler [2] proved that elements of this set have two other representations. One representation is of transfer functions of a certain type of unitary operator. The other representation is of contractive functions over ^-tuples of commuting strict contractions acting on a Hilbert space. We present and discuss J. Agler's results in this chapter. Furthermore, we prove transfer functions of a certain type of contraction operator are in the set Td{£-,£*)- 2.1 Representation and Realization on B_ Let £ be a Hilbert space and £(£) be the space of bounded linear operators from £ into £. An operator valued mapping I : ft X ft -» £(£) is positive-semidefinite if there exists an auxiliary Hilbert space M and a function H : tt —> C(M,£) such that l(z,w) = H(z)H{w)*. Let £ and £» be two Hilbert spaces and £(£,£*) be the space of bounded, linear operators from £ into £,. Define jF d (£,^) to be the set of £(£,£* )-valued functions defined on ID^ that have the factorization property TVd'- there exist d positive-semidefinite holomorphic operator mappings Ik : 1D^ x W~ — > £(£«) such that d TVd:l- W(z)W(wY = Y -(*,-, wfilfa w). *— ' s j=i ^(£,£'») is commonly called the Schur class. Functions in Fd(£,£*) have a transfer function representation. A d-variable unitary operator colligation S is a tuple S = (U.'H, £,£,), where "H,£, £* are Hilbert spaces, H, = ®j =l Hj has a fixed d-fold orthogonal decomposition, and U is a unitary operator u = A B C D The transfer function of the d- variable unitary operator colligation S is defined to be the operator-valued function defined on W Wx(z) = D + C(I- Z(z)Ay 1 Z(z)B where Z : -> U is defined by Z(z) = ®Ui Z i( z i) and Zi : B -> U % is defined by Z l (z) = zl n ,. Theorem 2.1.1 [2] Let £, £, be a Hilbert spaces. If f is a holomorphic €(£,£*)- valued function defined on W , then f G 7 d {£,£*) if and only if there exists a d- variable unitary operator colligation S such that f = Wy. ■ The representation found in Theorem 2.1.1 is a unitary realization of /. Func- tions in T&{£,£*) also have a representation as bounded functions over (/-tuples of commuting strict contractions acting on a Hilbert space. Let T = (7\, . . . , Tj) be any collection of d commuting strict contractions defined on a Hilbert space H and / be analytic £(£,£»)- valued function defined on 0*. The operator /(Ti, . ..,lrf) can be defined via the Cauchy integral formula having values in C{£ ® //", £» <8> H). Equiv- alently f(Ti,...,Td) can be defined via a power series expansion f(Ti,...,Td) = Efc c ( k ) ® T t ■ ■ ■ T d d where * = (*li • • • , fa) ranges over N d . Theorem 2.1.2 [2] Let £ and £* be Hilbert spaces. If f is a holomorphic £(£,£*)- valued function defined on 0, then f 6 J^(£,£«) if and only if ||/(T)|| op < 1 for every d tuple T = (7\, . . . , T&) of strict commuting contractions defined on a Hilbert space H . The Nagy Dilation Theorem and Ando's Theorem [24] can be used to show Td{£-,£*) is identical to the unit ball BH°°(V L ) for d=l,2 respectively. Hence Theo- rem 2.1.2 gives the equivalence between J-d{£, £*) and contractive analytic functions 'A B C D H S -¥ on the disk and bidisk respectively. For d > 2, Fi{£,E*) is properly contained in 2.2 Transfer Function Embedding Theorem Theorem 2.1.1 can be extended to transfer functions of contraction operators. Define a d- variable contractive operator colligation £ to be a tuple £ = ((/, %,£, £*), where "H, £, £» are Hilbert spaces, H = ® d j=1 Hj has a d-fold orthogonal decomposition, and U is a contraction operator U = The transfer function of the <f-variable contractive operator colligation £ is defined to be the operator-valued function defined on IF W L {z) = D + C{I- Z{z)A)- l Z{z)B where Z : O^ -► "H is defined by Z(z) = ® d l=l Z l {z l ) and Z t : D -4 % is defined by Theorem 2.2.1 (Transfer Function Embedding) // £ w a d-variable contrac- tive operator colligation, then there exists a 2d+l -variable unitary operator colligation T such that W r {z l ,...,z dy 0,...,Q) = Wz{z l ,... ) z d ). Proof: Assume that / = W% for some d-variable contraction operator colligation £. Extend U to a unitary operator using the rotation matrix Ru [14]: Ru = U {I-UU*)t {I-U*U)2 -u* H U £ -+ £* Q Q* where Q is the closure of the range of (I — UU*)? and Cj, is the closure of the range of (/- -U*U)$. Ru can be written in the following block form \A B Ai B x ' Ru = C D C x D x A 2 B 2 -A* -C* C 2 D 2 -B* -D* Using elementary column and row operations on R v we define a unitary operator Q: Label Q = A Bi A x B C 2 -D* -C* D 2 A 2 -B* -A* B 2 C C 2 C x D ~n~ H~ G -* Q* e £* A = A Bx Ax C 2 -D* -C* A 2 -B* -A* B B D 2 B 2 c=[c c 2 c x ] . Using Theorem 2.1.1 and Theorem 2.1.2 we conclude W(z) = W{zx,...,z d ,...,z 2d+ x) = D + C(I-Z(z)A)- l B is in T 2 d+x{£, £>)• Let T = (7\, . . . , Tj) be any rf-tuple of commuting strict contraction operators acting on a Hilbert space H. Computation shows W(Tx, . . . , Tj, 0, . . . , 0) = W(T u ...,Ti). Hence ||^(r 1 ,...,r rf ,o,...,o)|U = ||^(r 1 ,...,T rf )|| op <i, since (Ti, . . . , Tj, 0, . . . , 0) is a 2d + 1 tuple of commuting strict contraction operators. Conclude \\W(Tx,...,T d )\\ op <l for all (f-tuples of commuting strict contractions acting on a Hilbert space H. Using Theorem 2.1.2 conclude W € J-d{£,£*)- ■ Corollary 2.2.2 Let £,£* be Hilbert spaces. If f is a holomorphic £(£,£») -valued function defined on ID^, then f 6 J 7 d{£,£*) if and only if there exists a d-variable contractive operator colligation E such that f = W^.- 9 Proof: If / G Td{S,S„), then Theorem 2.1.1 shows that there exists a d-variable contractive operator colligation S such that / = W^. The Transfer Functions Em- bedding Theorem provides the converse. ■ We call Theorem 2.2.1 the Transfer Function Embedding Theorem. This name is appropriate due because we embed the transfer function of a (/-variable contraction operator colligation into a transfer function of a 2d + 1-variable unitary operator colligation. We embed a transfer function by doubling the dimension. CHAPTER 3 REALIZATION AND REPRESENTATION ON A POLYDOMAIN In the last chapter we found that a function / € Td(£,£*) if an d only if / = Wy. where £ is a d-variable unitary operator colligation if and only if / is a contractive function over d-tuples of commuting strict contractions. If we replace each occurrence of the Szego kernel in TVd with a suitable kernel kj defined on a region $7y, we obtain parallel results. 3.1 Realization on fi - Hi x • • • x Q^ For j = 1,2, .. . , d, let kj(z,w) be any positive kernel defined on a region fij for which the reciprocal kernel has one positive square, i.e. ±(z,w) = l-'£bj(z)bi> l (w) for some scalar valued functions b\,...,b J L defined on fl r The kernel defined as k = ki(zi,Wi)k2(z2,W2) . . . kd(zd, Wd) is a positive kernel defined on the polyregion Q = tti x ■ • • x Od. Given Hilbert spaces £,£*, let .F,fc(£,£*) denote the set of £(£,£*) valued functions W(z) = W(zi,Z2,...,Zd) which are defined in the polyregion (I = fix X • • • xfld and which satisfy the factorization property TVk'- there exist auxiliary Hilbert spaces Ci, . . . ,Cd and functions ffi, . . . , H& defined on U with values in C(Cj,£») such that TV k : h. - w(z)w( w y = T Uzj^H^Hjiwy. Kj For kj = j3-=, the Szego kernel, TVk corresponds to the outgoing kernel representation examined by Ball and Trent [9]. For d=l and b) (z) general for j = 10 11 1, . . . , Li, Tk{S, £„) corresponds to the unit ball of multipliers on the space H(k) for the NP kernel k [10]. Let L = (Li, . . . ,Ld) and define a d- variable L-ball operator colligation, E, to be a tuple £ = (U,H,S,S.), where ff,£,£, are Hilbert spaces, H = ®j =l H J has a fixed d-fold orthogonal decomposition, and U is a unitary operator where H' - ®j =1 [®i' Hj]. Let H'- = ©f J Hj and let Zj denote the analytic function defined on B> L > with values in C{H'-, Hj) defined by Define the polyregion B L = B Ll x • • • xB L « and Z : B L i-> £(#', #) by Zfc 1 , . . . , z d ) = ©fZj(z J ). The transfer function for the colligation £ is Wz{z) = D + C(I- Z(z)A)- i Z(z)B. Define E : U — ► hi as the embedding of Q into B^, given by E(z) = E(z u ...,z d ) = ((b\(z 1 ),...,bl(z l )),...,(bt(z d ),...,bl(z d ))). With this terminology we give the following theorem. Theorem 3.1.1 The £(£,£») -valued function W(z) = W{z\,. . . ,z d ) defined on ft — Qi x • • ■ x Qd is in the class Tk{£-,£*) if and only if there exists a d-variable L-ball operator colligation £ such that W — W^(E(z)). Proof: Suppose first that W(z) = W^,{E{z)) for a d-variable L-ball operator colligation £. Using the fact that Zj(E(zj))Zj(E(wj))* — 1 — y(zj,Wj), compute h. - W(z)W(wY = ^l(z i ,«; i )^ i (^ i H*, where Hj(z) : Hj -¥ £, is given by Hj(z) = C{I - Z(E{z))A)- l \ Hj . 12 Conversely suppose that h. - W(z)W(wY = J] i(*i,Wi)ffi(*i)J5ri(wir, We rewrite this identity in the form ^[ZAEiz^ZAEiw^TH^H^wr} + I = Y / H J (z)H J (wr + W(z)W(wy. (3.1) Let T* be the linear span of the functions Z x (E{w x )YH x {wY e» : w = (w x , . . . , i«d) e!l,e, G £* Z d (£(^))*#,H* / and J- be the linear span of the functions > c{® d k=l (®i k C k ))®£* H d (w)* W(w)* e» : w = (toi, . . . ,u>d) € ft, e» G £* > c(©LiCjb)ef. As a consequence of (3.1) there exists a well-defined linear isometry V from T onto «?■"» such that V: H d (wf W{w)* (- X Z x {E{w x )YH x {wY Z d (E{w d )YH d (wY I (3.2) for all e* £ £* and u> € ft. Choose Hilbert spaces Hk containing Ck such that dim(Hk Ck) = oo. It follows there exists a unitary map U : (® d k=1 Hk) (B£^ (®J»i(©i*&)) © fi. extending V. Set i/ = ®fHk and write (/ as a 2 x 2 block matrix (7 = A B C D 13 Let us set H(w) = ®fH k considered as an element of £(£»,#). Since U extends V from (3.2) and Z(E(w))* = ®iZ k (E(w k ))* we see that A* C* B* D* Z(E{w))*H{w)* I t » = H{w)* W(w)* e, for e, £ ^,. This generates the following system of operator equations: A*Z(E{w))*H{w)* + C' = H(wY B*Z{E{w))*H(iv)* + D* = W{wY From the first equation we solve for H(w)* to get H{w)* = (J - A*Z(E(w))*) 1 C*. Substituting this into the second equation yields or equivalently W(w) m = B*Z{E{w))*{I - A*Z{E(w))*)- l C* + D W(z) = D + C{I- Z(E{z))A)- l Z{E{z))B. We conclude W has the desired form with U. ■ 3.2 Representation on fl = f?i x ■ ■ • x Qj Define the set Tl(£,£*) to be the set of C(£ ,£„)-valued functions W such that W has the factorization property TV k with the kernels k 3 : B ' — >■ C defined as kJz.w) = i — — -*— — . This kernel is commonly known as the row contraction kernel [8] and in view of the preceeding theorem, if W G ^l(£, 5,), then H^z) = ^(z) for some d variable L-Ball operator colligation. Notice, if Li — 1 for all i — 1, . . . ,d we obtain J-l(£,£*) = J-d(£,£*)- The set Ti\£, £*) is of particular importance and we wish to characterize it in terms of bounded functions over the domain of some class of operators acting on a Hilbert space H . Let n be a positive integer and T = (Ti, . . . ,T n ) be an n tuple of operators acting on a Hilbert space H. We say T = (7\, . . . ,T„) is a strict row contraction if X^Li T*Ti < 1. Define the class 1Z to be the collection of d tuples of operators, T = (Ti, . . . , 7j), where Tj- is an L, tuple of operators that forms a strict row contraction. 14 Theorem 3.2.1 IfW is a £(£,£,) -valued holomorphic function defined on M L , then W <= T L {£,£*) if and only if\\W{T)\\ op < 1 for all T 6 K. Let Tell, then W(T) € C(£ , 5.) £(#) is defined by W{T) = ^2c(m)®T m 771 where VK(z) = £ TO c(m)z m , 2 € B L . More generally, if /i is a £(£, £,)-valued function defined on B^ x Ml holomorphic in the first variable and conjugate holomorphic in the second variable , then h(T) € £(£,£*) <g> C(H) is defined by / l (T) = J]c(m,n)®T* n T rn where h(z,w) = ^2 mn c(m,n)w n z m , z,w G B L . Theorem 3.1.2 follows from the following Theorem by letting h(z,w) = Is, - W(z)W(w)*. Theorem 3.2.2 Let C be a separable Hilbert space; if h = h(z,w) is a C(C)-valued function defined on Ml x Ml holomorphic in the first variable and conjugate holomor- phic in the second variable, then h{T) > for all T € n if and only if there exist d auxiliary Hilbert spaces M. r and d holomorphic C{M r ,C) valued maps f r r = 1, . . . ,d respectively defined on Ml such that d TV L ■■ k(z,w) = 5^( 1- < Zr,W r > C lr)fr(z)fr(wY r=l for all z,w E B^. Proof: First assume h has the form in TVl- Fix T 6 TZ. It follows h(T) = Eti fr(T){l-Zti «<t»n/ P (T)*. Since T ell, 1 -Ef=i «<«*' > and we conclude h(T) >0. To prove the converse direction fix a basis {e t } of C. Let H. denote the topo- logical vector space of holomorphic C(C) -valued functions defined on Ml x Bj, with the topology induced by the family of seminorms \\h\\ n = max e _^.^\\P n h{z,w)P n \\ ' ^n+1 •* 15 where -((-^-)B L ')x-.-x((^-)B L <), n + l n + l 7 ' n + l (-—)M L ' = {(z u ...,z Li ): zt e and £ \z t \ 2 < -2-}. n + l ^"^ r» t 1 Let P n denote the orthogonal projection of C onto the span of {e, : 1 < i < n}. The topological vector space H carries a locally convex Hausdorff topology. Let O C H denote the set of all h € U such that h(z,w) = Er=i( 1_ < z n w r >c^ )fr(z,w) where f r is positive semidefinite holomorphic on Bj, x Bl for each r. The fact that O is a convex cone is easily verified. We claim O is closed in H. To see this assume that h j (z,w) = Er=i(!- < z >-, w <- >&r)fr{z,w) € and ^(2,t«) -> &(*,«;). For n > 1 inductively construct a sequence {#}, {j'J }, . . . as follows. Since ||/* J ||i -► ||/i||i, we have Km^J^l- < z r ,z r >)P 1 ft(z,z)P 1 = Pihiz y z)Pi for z £ |B/,. Since ft is positive semidefinite, f?(z,z) > 0, and since ||/i J ||i forms a bounded sequence there exists a positive bounded function g : |B^ — > C(P\C) such that Pji(Z:Z)Pl<9(z) for all j ,r, and 2 6 §B/,. Finally, since /■? is holomorphic positive semidefinite we can conclude there exists a subsequence {j/} and d C(P\C) valued positive semidefinite holomorphic functions gl(z,w) such that P l f^(z,w)P 1 ^g 1 r (z,w) uniformly on compact subsets of |B^ x |B£ . Now suppose that sequences {jf },..., {j" -1 }; {jf} C {jf -1 }, have been de- fined with properties Pif?(z,w)Pi^gi.(z,w) 16 for i = 1, . . . ,n- 1. The argument in the preceding paragraph shows that there exist a subsequence {jj 1 } of {j,"" 1 } and d C{P n C) valued positive semidefinite holomorphic functions g™(z,w) such that P n f?{z,w)P n -*g?(z,w) uniformly on compact subsets of ^jBl x ^-Bl Now define d holomorphic I 2 matrix-valued functions G r (z, w) on B L x B L by the formula G r (2,u;)ij = /imn-^oo < g?(z,w)ej,ei > G r is well defined by construction: if m < n, then g™{z,w) = P m g?(z,w)P m . Fur- thermore since by construction, < h{z,w)e j ,e l >= ]T](1- < Zr,w F >)G r {z,w) i: > (3.3) provided that there exist C(C) valued holomorphic maps g r with < g r (z,w)ej,ei>=G r {z,w) (3.4) (i.e. G r (z,w) is a bounded operator on I 2 ). Since G r (z,z) is positive semidefinite, (3.3) implies that G r (z,z) is bounded. Since G r (z,w) is positive semidefinite G r (z, to) is bounded. Hence (3.4) defines C(C) valued maps g r and we obtain from (3.3) h(z,w) = 2^(1- < «,«; >)flf r (z,u>) which establishes that O is closed. Now assume that h is a holomorphic C(C) valued function on El x Ml with the property that h (T) > for all T £ 11. Let us show h € O. Since is closed, the Hahn-Banach separation principle implies that h £ O if and only if ReL (h o ) > (3.5) 17 whenever L G "H* has the property that ReLo(h) > for all h G O. (3.6) Assume that L 6 K* and that (3.6) holds. We must show (3.5) holds. For h G H define /j v (z,u;) = fc(t»,z)'. Define E € W by the formula 1 E(*) = 2(Io(*) + W))- Let "H denote the vector space of holomorphic C(C,C) valued maps defined on M L . Define a sesquilinear form [ , ] on H by the formula [f,9] = E(f(z)g(wy). Observe that if h G K and h - /i v then £(/*) = ReL Q {h). Hence since {f{z)f{w)*) v - f{z)f{w)* for all / G "H we deduce from (3.6) and the fact that f(z)f(w)* G C* that [ , ] is positive semidefinite on % . Letting N = {f 6 % : [/, /] = 0} we deduce via Cauchy's integral formula that AMs a subspace of Ho and that [ , ] induces an inner product on jf. Let W. 2 (E) denote the Hilbert space obtained by completing jf with respect to this inner product. Densely define Li + ■ ■ ■ + L d operators T = {{M-} i J zl } d j=1 acting on jf- by the formula (m!/)(z) = 4m z = (zi,...,z d ) G M L z. t = (z\,...,z[ t )e >Li Fix / G n 2 {E) and i G {1, . . .,<*}. We have ni ( E ) -\\(Mlf,...,MiJ)\\^ in2(E) = E(f(z)f(wY)- < z u wi > c l, E(f(z)f(wy) = E{(l-<z i ,w i > c i i )f(z)f{wY)>0, 18 since (1- < z i ,w l > c t i )f(z)f(w)* 6 O. Hence T is not only well denned on ^f but extends by continuity to a contraction defined on H 2 (£). Conclude pT e H for p < 1. Fix h = £ mn c m „^ m € H, let />< 1, and let / = £ e, 0/^C® H 2 (£). Let us derive a formula for < h(pT)f, f >. < h( P T)fJ>= Y / < C mn® (pTy n (pT) m fJ> mn = E E < c - ® (prripTrie, ® /,), e ,- ® /,- > mn ij = E E < ** e i' e « > iwvfhiflTft] mn ij = EE < Cm " e ^ e ' > *((W"M~/i(*)/«(wr) = £ I E < Kp z ,P w ) e n e J > fj( z )fi( w Y ) ■ Letting / = Y^=i e j ® ( le i) m tne aDove calculation where (lej) is the natural embedding of the mapping 2 -» zej € £(C,C) in U 2 {E), we find [Mpr),/,/] = £?(PnM^»/>w)i 5 »). Since pT eK , [h (pT)JJ] > for all /0 < 1. Hence E(P n h o (pz,pw)P n )>0 whenever p < 1 and n > 1. Fix p < 1 and let n -> 00, by the continuity of £ we deduce that E(h o {pz,pw)>0 (3.7) for all p < 1. Letting p — > 1~ and using the continuity of £ we deduce E{h ) > 0. (3.8) Finally, since h (T) > whenever T € 72. we claim that /i = /i^ . To see this take T = (21 , . . . , Zd) € B>l . In particular, we have that E(h ) = ReL{h ) and (3.8) implies that (3.5) holds. This establishes the theorem. ■ CHAPTER 4 INTERPOLATION The classical Nevanlinna-Pick interpolation theorem states given n distinct points z 1 , . . . , z n in D and n points y 1 , . . . , y n in C, there exists / G ##°°(D) such that f(z l ) — y l for i = 1, . . . , n if and only if the associated Pick matrix [(l-i«!SH*,*i)]y (4-0) is positive-semidefinite. This theorem was extended to include the generalized Schur class Fd{£,£*) as the interpolating set by J.Agler [1]. In this chapter we present the Nevanlinna-Pick interpolation theorem for ^(5, £"♦). Using the results found in chapter three, we prove our version of the Nevanlinna-Pick interpolation theorem with J-k(£,£*) as the interpolating set. 4.1 Interpolation on fgj To formulate J. Agler's Nevanlinna-Pick interpolation theorem, let z 1 = (z\, . . . ,z\), . . . ,z n = (z", . . . ,z%) be n distinct points in U*, Mi, . . . ,M n be n auxiliary Hilbert spaces, x t , . . . , x n be n operators in £(£"», -Mj) respectively, and j/i, . . . , y n be n operators in £(£,Adj) respectively. The associated interpolation problem is /: Is there a W € J~d{£i £*) such that XiW(zi) = j/j- for t = 1, . . . , n. Theorem 4.1.1 (Agler's Nevanlinna-Pick Interpolation Theorem) [2] [9] [3] Let {z , . . .,2 n ,£i,. . . ,x n ,yi,. . . ,y n ] be an interpolation data set as shown above. Then I has a solution if and only if there exist d positive semidefinite n x n block matrices M l = [M\-] such that *i-ra£ = ET^K /=i 1!) 20 In the case E = £ m = Qd=l, and z, = 1 for i = 1, . . . , n it is easily seen that Theorem 4.1.1 reduces to the classical Nevanlinna-Pick Interpolation Theorem. For d = 2, the set JF 2 (C,C) is the ball in H°°{H?) and hence Theorem 4.1.1 gives a necessary and sufficient condition for interpolation in the set BH 00 ^) (the unit ball in H°°(&)). For the case d > 2, ?*{£,£*) is properly contained in the set BH 00 ^) (the unit ball of the space of bounded analytic C(6, £»)-valued functions defined on D**). Thus the condition presented in Theorem 4.1.1 is in general sufficient but not necessary for interpolation in the set BH (X '{W). 4.2 Interpolation on f? = Oi x ■ ■ ■ x flj Using results found in chapter three we can prove a Nevanlinna-Pick inter- polation theorem with ^-(5,5*) as the interpolating set. Given n distinct points z l = {z{,...,z$),...,z* = ( z iT--, z d) inft = fli X •••Xftrf, n auxiliary Hilbert spaces Mi, . . ■ ,M n , n operators Xi, . . . ,x n in C(€m,Mj) respectively, and n opera- tors yi, . . . , y n in C(£, Mj) respectively. The associated interpolation problem is I: Find a W € Tk{£, £*) such that XiW{zi) - y; for i = 1, . . . , n. Theorem 4.2.1 Let {z 1 ,. . . ,z n ,Xi,. . .,x n ,yi,. . . ,y n } be a interpolation data set as shown above. Then I has a solution if and only if there exist d positive semidefinite n x n block matrices M l = [M 1 -] such that d **% ~ ViVj = £ r(^W)4 (4-1) /=i ' Proof: Suppose that W £ Tk{£,£*) satisfies the interpolation condition X. Hence by definition we know that there exists holomorphic functions Hj(z) such that d is. - w{J)W{iiy = £ T (z},z dh^h^)*. Hence - 1 xi(h. - W(z*)W(z>Y)x* = J2 ■j-(zj,z 3 l )x i H l (z i )H l (zi)*x* i . i=i ' 21 Using the interpolation condition (xiW(z % ) = y,) we see (4.1) holds with rt ii = x i Hi{f)H l {z>Yx). If we let M l be the block matrix [My] it is clear from the form of the My that M' is positive semidefinite. Now suppose that there exists positive semidefinite matrices M 1 , . . . , M d for which (4.1) holds. As each M l is positive semidefinite, we may factor M l as M l = A 1 (A 1 )* where A 1 = £(C/,©"X ! ) for an auxiliary Hilbert space Ci , I = 1,... ,d, and let |"41 fhereA(€£(a,M)- A WJ Using the fact that Zj(E(zj))Zj(E(w ] ))* — 1 - -^(zj,Wj), we can rewrite (4.1) as ^(^))z,(j5(*/)ri4i(A5n + ««*; = 5] 4(4r + wff- ( 4 - 2 ) /=i /=i Let Q % be the span of the set of elements Z x {E{z\)nA^ z^E^mAfr X ; m 3 : rrij € A4j,j = 1,.. .,n > C (©Lil©!^)) © £ * And let (/ be the span of the set of elements [AJT y 3 . m 2 : rrij e Mj,j = l,...,n> C {@t=i c k))@ €■ As a consequence of (4.2) there exists a well-defined linear isometry V from Q onto G* such that I" i\* (4 (A?)' ?«, Z 1 ( J E(^))*(^ 1 ) cf\* Z„(tf(aj))'MJ) x j m, (4.3) 22 for all rrij € Mj and j = 1, . . . , n. Choose Hilbert spaces H k containing C k such that dim(Hk 6 Ck) = 00, It follows there exists a unitary U : {® d k=1 H k ) © £ -> (® d k=1 (®i k Hk)) © £. extending V. Set # = ® d l H k , write ^ = A B C D in 2 x 2 block operator notation, and set Z{E{z)) = ®tZ l {E{z l )) for x = {z u . . . , z d ) e Q. Define W(z) = D + C{I - Z(E{z))A)- l Z{E{z))B. Then VK G :F*(£, £,) by Theorem 3.1.1. To show W satisfies the interpolation condition X we proceed in a manner similar to the last part of the proof of Theorem 3.1.1. For a fixed j = 1, . . . ,n let Hj be the operator ff; = (A?)' considered as an element of £(A4j, #). Since U extends V we know by (4.3) that A*Z{E{z i )) m H i + C'x* = Hj B'Z{E{zi)YH 3 + D*x* = y*. Solve the first equation to obtain ^ = (I - A*Z{E{z>)YY*C*Xj Plugging this into the second equation we obtain (B*Z{E(z j ))*(I - A*Z(E(zi))*)- l C* + D*)x* = y*. Taking the adjoint of both sides we obtain the desired result XjW(z^) = y r ■ CHAPTER 5 CORONA THEOREMS Suppose that a u . . . , a n are complex valued functions in i/°°(D). The Carleson Corona Theorem [11] asserts the existence of /i,. . . ,/« solving £?_, /<** = 1 if and only if there exists a 5 > such that inf lz{<1 {\ ai (z)\ + --- + \a n (z)\}>6>0. The Toeplitz Corona Theorem [22, 17] states that there exist functions f t G ff°°(D) for j = l,...,rf such that £" =1 /« a « = ! and stt PW<i{ELl l/'( 2 )| 2 } < £ if and only if T a X^--- + T an T: n -j 2 I>Q (5.1) where T ak : fc(z) -> afc(z)/i(2) is the analytic Toeplitz operator on the Hardy space H 2 (D) with symbol a*. Condition (5.1) can be expressed as E a l {z i )a l {z j ) + h a n {zi)a n {zj) - 8 2 _^ > l - z;*7 ° tCj - for all complex numbers Ci,-. . . ,cjy and all points Zi, . . . ,2jv € D for A 7 = 1, 2, 3, In this chapter we present two versions of the Toeplitz Corona Theorem. The first version was formulated for functions defined on the polydisk IF [9]. Using the results in chapter three we prove the second version for functions defined on the poly region f2. 5.1 Toeplitz Corona Theorem on W Ball and Trent [9] formulated a version of the Toeplitz Corona Theorem for the polydisk P* . 23 24 Theorem 5.1.1 [9] Let a u ...,a n be complex valued functions in H 00 ^) and let 8 be a positive number. There exist functions /i,...,/ n such that the column matrix function [/i ... f n } T is in the set ^(C, C") and ai {z)f x {z) + ■■■ + a n (z)f n (z) = 1 on W- if and only if there exist auxiliary Hilbert spaces &,. . . ,Cd and d holomorphic functions Hi(z), . . . , H d {z) on Bf 1 , with H k (z) having values in £(C k ,C) such that n ^ 1 Y, a k {z)vM - S 2 = J2 ~( z k, w k )H k {z)H k (w)* k=\ fe=i for all z,w G &. The techniques found in Ball and Trent lead them to a more general theorem. Theorem 5.1.2 Let £\,£i,£z be three Hilbert spaces and suppose that A and B are given bounded holomorphic functions on W with values in £(£2, £3) and £(£1,^3) respectively. Then there exists a F G T&{£\,£i) with A(z)F{z) — B{z) on B 1 if and only if there exist d auxiliary Hilbert spaces C\, . . . ,Cd and d holomorphic functions Hi(z), . . . , Hd(z) on Bf 1 , with H k (z) having values in C(C k ,£ 3 ) for k = 1, . . . ,d, such that - 1 A(z)A(wy - B(z)B(w)* = Y -(z k , w k )H k (z)H k (w)* for all z,w 6 0*. s k=i Notice we recover Theorem 5.1.1 from Theorem 5.1.2 by taking 6\ = £ 2 = C, £ 2 = C\ A(z) = [ ai (z) . . . a n {z)l and B(z) = 6. 5.2 Toeplitz Corona Theorem on = Oi x • ■ • x fid Let £ and £* be two Hilbert spaces, k\, . . . , kd be positive kernels defined on Oj whose reciprocal has only one positive square. Using results found in chapter three we prove an operator version of the Toeplitz Corona Theorem for the kernel k{z,w) = ki(zi,wi) . . . kd(zd,u>d) defined on the polyregion ft = Qi x • • • x fid- 25 Theorem 5.2.1 Let £i,£ 2 ,£ 3 be three Hilbert spaces and suppose that A and B are given bounded holomorphic functions on ft with values in £(£ 2 ,£ 3 ) and £(£i,£s) re- spectively. Then there exists aF e ^(£i,£ 2 ) with A(z)F(z) = B(z) on ft if and only if there exist d auxiliary Hilbert spaces Ci,...,Cd and C(Ck,S 3 ) -valued holomorphic functions Hk(z) defined on ft for k = 1, . . . ,d such that d d A{z)A(w)* - B(z)B(wY = Y, -j-(z h wi)Hi(z)Hi{wy for all z,w € ft. Notice we recover Theorem 5.1.2 from Theorem 5.2.1 by letting £\ = £3 = £*, £ 2 = 0?£, A(z) = [ai(z) ... a n (z)], B{z) = SI £m , and fc, = s for / = l,...,d. Proof: Suppose that there exists a F(z) € Tk{£u&i) such that A(z)F(z) — B(z) on ft. Using Theorem 3.1.1 we know that there exists a c/-variable L-Ball operator colligation S such that F(z) = Wz(E(z)), where z -+ E(z) is the embedding of ft into B Ll x • • • x B Ld defined in Theorem 3.1.1. From A(z)F(z) = B(z) and using the fact Zj(E(zj))Zj(E(wj))* = 1 - ~(z J: Wj) we deduce A(z)A{wj* - B(z)B{w)* = A(z)(I - F{z)F(w)*)A(w)* Kj W here H,(z) : H 3 -> £3 is given by Hj(z) = A(z)C{I - Z(E(z))A)- l \ Hj . Conversely suppose that d - 1 A{z)A(w)* - B(z)B(wy = Y, r(*«' m)Hi(z)Hi(wy. 1=1 We rewrite this identity in the form ^LZiiEizMZjiEMYHiWHAwy] + A(z)A(w)* = Y / H J (z)H ] (wy + B(z)B(wr. (5.2) 26 Let T* be the linear span of the functions Z 1 (E{w l )yH 1 (wY Z d (E(w d ))*H d (wy A(w)* e. : to = (wi, . . . ,w d ) € n,e* € t+ \ c (eLil©^)) © £* and .F be the linear span of the functions AM' e» : to = (u>i, . . . , w d ) € fl,e» € £ .1 C(®i =1 C k )@€. As a consequence of (5.2) there exists a well-defined linear isometry V from T onto T* such that V H^wy HdH* B( w y 'ZiiEfaWHtW z d (E(w d )yH d (wy A( w y (5.3) for all e» € £». Choose Hilbert spaces H k containing Ck such that dim(H k QC k ) = 00. It follows there exists a unitary map d (r&L k U : m =l H k ) ® £ ^ (®Li(©i #*)) © ^. extending V. Set H = Q)fH k and write /7 as a 2 x 2 block matrix [/ = A B C D Let us set H(w ) = ($)fH k (w) considered as an element of £(£3, #). Since £/ extends V from (5.3) and Z(E(w))* = ®\Z k {E{w k ))" we see that A' C* B* D* z(E{w)yH{ w y A{ w y e. = B{ w y e, for e* G £, This generates the following system of operator equations: A*Z{E(w)yH( w y + C*A{w)* = H(wY B*z{E(w)yH(wy + D*A{ w y = B( w y 27 From the first equation we solve for H(w)* to yield H{io)* = (I - A*Z{E(w)yr l C*A{w)*. Substituting this into the second equation yields B(w)* = B*Z{E{w)Y(I - A*Z{E(w)Y)- l C*A{wy + D* A{w)* = (B*Z{E(w)) m (I - A*Z{E{w)Y)- l C* + D*)A{w)* or equivalently B(z) = A(z)F(z) (5.4) where F(z) = D + C{I - Z{E{z))A)- 1 Z(E(z))B. Using Theorem 3.1.1 we know F £ Tk{£li&l) an d (5.4) gives us our desired result. ■ CHAPTER 6 SYSTEMS THEORY 6.1 Roesser Model Define a d- variable operator colligation X to be a tuple X = (U,7i, £,£*), where W, S, £, are Hilbert spaces, U = ® d j=l Hj has a fixed d-fold orthogonal decomposition, and U is a bounded operator U : % ® £ ^ % ® £* (6.1) U = A B C D H % is known as the state space, £ is known as the input space, £, is known as the output space, and U is known as the connecting operator. As we have mentioned before, the colligation is unitary or contractive according to whether the connecting operator is unitary or contractive. The transfer function of the d-variable operator colligation S is defined to be the operator-valued function defined on Tr W E (z) = D + C(I - Z(z)A)- l Z{z)B where Z : D* -> U is defined by Z(z) = ®f =1 Z t {zi) and Z { : D -> Hi is defined by Zi(z) = zI Hi . Associated with any of-variable operator colligation is a ^-dimensional discrete- time linear system. The time variable n for this system is a d -tuple n = (f»i, . . . , Hi) of integers. Define Uk : Z d ->■ Z d to be the forward shift in the k th coordinate a k {n ll ... J n d ) = (n u . . . ,n k + 1, . . . ,n d ). The input-state-output linear system associated with (6.1) can be written as the following system of equations (6.2) A B C D x{n) u(n) x(e(n)) y(n) 28 29 where u(n) is the input signal with values in £, x(n) = xi(<ri(n)) is the state vec- xi(n) , and y(n) is the output tor with Xk having values in Wk, x(a(n)) x d ((T d (n))_ signal with values in £». Form (6.2) is referred to as the Roesser Model [15, 16] in multidimensional systems theory. Let us motivate why the Roesser Model is studied. The input to output characteristics of a multidimensional linear time invariant system is modeled by a multivariable function H{zu .■•,*«*)■ This function is commonly called the transfer function of the system (this motivated our use of this terminology earlier). If we assume H(z) is a analytic function in some region fi C C which includes the origin (0, . . . , 0) then H(z) has a power series expansion H(z) fc=(O,...,0) (6.3) k _ „*i ■, k d valid in Vt where k = (fci, . . . , k d ), h € Z + , for i = l,...,d and z k = z x x ...z Physically, the indeterminates Z\,...,z d are the respective delay variables along the spatial or temporal directions of sampling during analog to digital conversion of a multidimensional spatio-temporal signal. Functions of the form found in (6.3) are commonly called of-dimensional filters. A realization of H{z) is a d-variable operator colligation S generating a linear system of equations found in (6.1) and (6.2) with Wy,{z) — H(z). Thus a Roesser Model is a input-state-output model of a realization S of a ^-dimensional filter Wz{z). We discuss two aspects of multidimensional systems: (1) the physical inter- pretation of the unitary or contractive properties of U as a energy conserving/energy dissipative property and (2) the realization and dimension of the state space Ti of a realization for a real valued rational function H(z) in 2 variables. 30 6.2 Energy Conservative /Energy Dissipative Linear Multidimensional Systems The results of chapter two show that the elements of the set Fd{£,£*) are transfer functions of d-variable contractive and/or unitary operator colligations. Let / : B* -* £(£,£,) be analytic and suppose there exists a d- variable unitary opera- tor colligation E = (U,H, £,£,) such that / = W?. Since U is isometric it follows \\U(h,e)\\ = ||(/i,e)ll for all (h,e) € U®£. In particular ||/i|| 2 + ||e|| 2 = \\U{h, e)\\ 2 . Physically, this is a energy conserving property. On the other hand, let / : Vr -> £(5,£») be analytic and suppose there exists a d-variable contraction operator col- ligation E = (U,H,£,£*) such that / = W%. Since U is a contraction it follows \\U(h,e)\\ < \\{h,e)\\ioiall{h,e)en®£. In particular \\U(k, e)|| 2 < \\h\\ 2 + \\e\\ 2 . Physically, this is a energy dissipative property. Thus functions in Td{£,£*) have realizations in terms of operators that have either energy dissipative and/or energy conserving properties. Moreover, functions that have either a energy dissipative and/or energy conserving realizations are in Fd{£,£*)- This gives the results found in chapter two physical meaning. In partic- ular the Transfer Function Embedding Theorem show that we can embed a transfer function of a d-variable contraction operator colligation in a transfer function of a 2d + 1-variable unitary operator colligation. Physically this can be viewed as a law of entropy for transfer functions. In particular, a energy dissipative system can be embedded into a larger energy conserving system. In this way, the results of chapter two can be viewed as empirical laws for multidimensional systems. 6.3 Minimal Realizations We call the dimension of the state space the order of the realization. Given H(zi,z 2 ) a rational 2-dimensional filter, the minimal realization problem asks does there exist a method to find a realization E of H such that the dimension of the state 31 space H is as small as possible, hence there exists no other realization A of H that has a smaller order. A method to develop such a realization in general is unresolved in the open literature, but methods exist for special cases: a) 2-d transfer functions with a sep- arable numerator or denominator [18]; b) 2-d, 3-d, and N-rf transfer functions that can be expanded into a continued fraction expansion [21, 19, 6, 20, 5]; c) 2-d all-pole and all-zero transfer functions [25]. To keep the notation of the literature, let us replace the indeterminates Z\,z% with zl 1 ,z^ x representing the delay elements. A general method producing a low order, but not necessarily minimum order, realization for a 2-d transfer function H(zi,z 2 ) was given in [18] in terms of hardware design of the delay elements z[ , z^ . To begin, we write the transfer function E{z x ,z 2 ) in terms of a rational function of 2.J 1 with polynomial coefficients in z± m , srvm.ta- 1 ) + • • • + *?M£) + /o(*r x ) ,,,, H{ ^ Z2) - z;"» qm2 (z; l ) + ~. + z? qi (z?) + qo (z; i y [ * } Without loss of generality, we can assume qoiz^ 1 ) = 1 + go^f 1 ). Let n to be the maximum degree of the polynomials f mi , . . . , f and q m2 , . . . , q . Then we can write fW ) = EL, fa? ^d q 3 {zT X ) = £*=o q\z- x k for i = 0, . . . , m, and j = 0, . . . , m 2 . The realization of H(z\,z 2 ) is shown in Fig. 6.1 and consists of three arrays of delay elements. One array consists of n elements of type z^ 1 are commonly shared to realize both numerator and denominator. The two other arrays consist of mi and m 2 elements of type z^ 1 that feedforward and feedback to realize numerator and denominator respectively. The order of the realization is n + mi + m 2 . This realization technique is a 2-d extension of the controller canonical form where the gains of the multipliers are functions in z£ . The drawback with this realization technique is that the feedforward delay elements of type z^ 1 and the feedback delay elements of type ^ 2 _1 are not shared. Indeed, this realization technique will not give the minimal realization in all examples. We will discuss examples later. 32 *? W o— 6— &-o ^0 L q •7? -o ►o- •o ■* ) ■*■ v ^ r ±r~^-' -H •»■ f °" X - o— ra— o ml -1 Z z delays A- -1 z, -O In *-( ■»■ delays <u, In ' ~£~^ m2 delays Figure 6.1: n + mi + m 2 -Order Realization [18] 33 As we have mentioned, the realization technique found in [18] will not always produce the minimal realization of a 2-d transfer function. The drawback is that the delay elements of type z? 1 are not shared. To improve this method, let us devise a method of sharing the delay elements zj 1 . Begin by writing down the transfer function H(zi,z 2 ) as shown in (6.3). Without loss of generality assume mi = m 2 . Then (6.3) induces two vectors A/*, V with polynomial entries M = v = fofr 1 ) q mi [zi ) *(*r x ) (6.4) (6.5) where M denotes the coefficients of the numerator of H and V denotes ^he coefficients Xi of the denominator of H. Define the shift S of a column vector x — x 7711 as s X\ Xi ■^rn\ ^TO] —1 Define the k projection Vk of a column vector x Xi '7711 as ~Xi~ Xi ^ Xk •^7711 .0. To share delay elements of type z 2 in the feedforward and feedback arrays shown in Fig. 6.1 we must find constants ai,...,a,k such that V k (V) = a x V k {N) + a 2 V k {S{M)) + ■■■ + a k V k {S k -\M)) (6.6) 34 r-r vy v^ *■ ( ) ■*" / ml -1 CP>A^^^O^^- o *( «- -o JL, z -o *K «- ts: 1-1 -O- ml ■ I delays -i ^JL 1 I 7.i m2 k I "■I o - »- -*- »- -«- n z,' delays 1 ^- 9 5 m2-k z,' delays Figure 6.2: n + m\ + m 2 — k -Order Realization for some fc = 1,2, In other words, the entries of V must be eliminated by the span of copies of the shifts of M . For each positive k for which (6.6) holds we feedback the feedforward delay elements of Fig. 6.1 as shown in Fig. 6.2. We bookkeep the sum of the feedback gains during this process and we subtract this sum from the denominator. We obtain the following matrix equation: V-Y^^S'-'M 1=1 l'm2-k\ z i ) w 1 ) J where /;(zf l ) = Yl™=i ^k z i f° r i = 0? • • • ? m 2 ~ k. We then realize this difference using m.2 — k delay elements of type z^ 1 as shown in Fig. 6.2. The order of the 35 realization is n + mi + m 2 - k. The same process can be applied with the roles of V and Af reversed. The difference is we feedforward the feedback delay elements of Fig. 6.1 in a similar method as shown in Fig 6.2 to produce a realization of order n + mi + m 2 — k. Starting from a transfer function H{z u z 2 ), our method provides a low order realization of H. A natural question arises: given a realization £ of H can we verify that £ is a minimal realization of HI This question is unresolved in the literature, nevertheless, we can derive a method that reduces the order of a realization. Let S = (U,%£,£*) be a d-variable operator colligation. Let P k : H -> H k denote the orthogonal projection. The colligation S is closely inner connected if the smallest subspace invariant for A,P u ...,P d and containing im B is the whole space H. Similarly, £ is closely outer connected if the smallest subspace invariant for A*,Pi,...,Pd and containing im C* is the whole space H. For one variable d = 1, closely inner connectivity is controllability. Similarly, for d = 1, closely outer connectivity is observability. Thus closely inner and outer connectivity can be viewed as extensions of controllability and observability. Ball and Trent [9] discuss closely inner connected and closely outer connected colligations in the context of isometric, coisometric, and unitary operator colligations. We will discuss these ideas in the context of reducing the state space of a realization. In [9], Ball and Trent point out that, in general, given a d- variable operator \A B\ colligation S = (U C D ,%,£,£*), the compressed colligation So = (U = PoA P B C D \y. ®£ i^-oi £,£*) where W C H is the smallest subspace invariant for A, Pi, . . . , Pd and containing im B (closely inner connected), and P : % — > Ho is the orthogonal projection, retains the same transfer function. In other words, Wz = W-£ ■ Similarly, if Ho is the smallest subspace invariant for A*,Pi, . . . ,Pd and containing im C* (closely outer connected) 36 then W^ = Wz . Hence if M is not both closely inner connected and closely outer connected, then we can project down to a smaller invariant subspace and obtain a realization of smaller order. This method combined with our previous results gives us an algorithm to obtain a low order realization of a 2-d transfer function H(z u z 2 ). ALGORITHM: 1. Write H(z u z 2 ) in the form given in (6.3). 2. Realize H according to Fig. 6.1. 3. Write down the numerator and denominator vectors J\f and V. 4. Check if equation (6.6) holds for any positive integer k. If so, then realize H according to Fig. 6.2. If not, keep realization found in step 2. 5. Label delay elements and write down the matrix ^ = r> n • 6. Check to see if the realization is closely outer or closely inner connected. If so, stop. If not, then project to smaller invariant subspace and repeat step 5. EXAMPLE 1: Let H 1 (z u z 2 ) = f^. We write H x {z ly z 2 ) = 'l^i • Ap- plying the method presented by [18] it is easily seen that H\ has a order 3 realization with circuit diagram shown in Fig. 6.3. Let us apply our technique. Writing the numerator and denominator vectors we obtain M = ' 1 " -1 *1 v = i Notice the first entry of M is 1 while the first element of V is (6.7) These two elements are linearly independent, thus we cannot find scalars ai,...,a& such that (6.6) will hold for any k. Hence we cannot use feedback to produce a lower order realization. u •O •o- •o- -o o t X 3 -o Figure 6.3: Order 3 Realization of H x 37 Writing down A, P, C, and D: -1 A = 1 T B = l C= [1 1 o] D = ( ). The orthogonal projections Pi and P 2 correspond to the labels on the delay elements in Fig. 6.3. In other words Pi corresponds to the state x\ hence it is the projection of C 3 onto the first coordinate. Similarly, P 2 corresponds to x 2 and x 3 , hence it is the projection of C 3 onto the second and third coordinates. It is easily verified that this realization is closely inner and outer connected. According to our algorithm, we stop. Conclude Hi has a order 3 realization. Indeed it is proved in [18] that 3 is the order of a minimal realization of Hi. 38 EXAMPLE 2: Let H 2 {z u z 2 ) = ..jV-i ,-■ • Applying the method pre- sented by [18] it is easily seen that H 2 has a order 6 realization with circuit diagram shown in Fig. 6.4. Let us apply our technique. Writing the numerator and denomi- nator vectors we obtain V K 2 1 VI ~i i _ (6.8) Notice that V\V = V\ Af. We feedback according to Fig 6.2 to obtain a realization of order 5 as shown in Fig. 6.5. Writing down A, B, C ', and D: A B C D = 0. -1 1 1 1 1 T () ll It is easily verified that this realization is closely inner and outer connected. Accord- ing to our algorithm, we stop. We conclude H 2 has a order 5 realization. To demonstrate how powerful the idea of closely inner and outer connectivity are let us show that we could have projected down the order 6 realization of H 2 shown in Fig 6.4 to obtain a order 5 realization. Writing down A, fi, C, and D for the realization shown in Fig 6.4 we obtain: A = B _; 1 1 1 1 [ (J 1 T I) C = [0 1 0] D = 0. 39 This realization is not closely inner connected. The smallest subspace invariant for A, Pi, and P 2 and containing im B is the space spanned by {ei, e 2 , ^( e 3 + es), e 4 , e 6 } where e,- is the column vector consisting of all zeros except for 1 in the i* entry. Projecting down on this invariant subspace we obtain a realization of order 5 with A, B, C, and D: ■o -1 1 A = y/2 1 1 i "1" B = C = [0 1 o] D = ( ). It is easily verified that this new realization is both closely inner and outer connected. •o- •o- 5a °x A X 3 o- 5: *q #• Figure 6.4: Order 6 Realization of H 2 40 •o- •o- 9x o. A o- •o l z 2 Figure 6.5: Order 5 Realization of #2 CHAPTER 7 CONCLUSION In chapter two we denned the Schur class jF d (£, £»). The results of J. Agler [2] showed elements of the set .F d (£,£«) have two other equivalent representations. One representation is of transfer functions of d-variable unitary operator colligations. The other representation is of contractive functions over (/-tuples of commuting strict contractions acting on a Hilbert space. Also included in chapter two was the Transfer Function Embedding Theorem. The Transfer Function Embedding Theorem can be viewed both mathematically and physically. Mathematically, transfer functions of d- variable contraction operator colligations are in the Schur class ^ r d(£, £*). Physically, it is a law of entropy. In chapter three, we generalized the results of J. Agler found in the previous chapter to ^ r / c (£,£*). Elements of Tk{S, £») are transfer functions of (/-variable L- ball operator colligations. Moreover, elements of ^jfc(£, £*) are contractive functions defined over a class of operators 1Z acting on a Hilbert space. Using the results in chapter three we proved versions of the Nevanlinna-Pick interpolation theorem and the Toeplitz corona theorem found in chapters four and five respectively. In chapter six, we discussed the minimal realization problem for a 2-d transfer function. As we have mentioned, this problem is still unresolved in the open literature. Moreover, due to the complexity of the problem, it is not a subject of active research today. Indeed, most of the research on this problem was done over two decades ago [18, 15, 16]. Hopefully, the methods and ideas found in this dissertation will lead to new insight on the problem. In particular, how are the ideas of inner and outer connectivity related to the minimal realization? We know we can reduce the ■II 42 order of a realization using inner and outer connectivity, but when does this method fail? Hopefully, these questions will lead to answers and new insights concerning the minimal realization problem. REFERENCES 1] J. 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Kailath New Results in 2-D Systems Theory, Part II: 2-D State- Space Models-Realization and the Notions of Controllability, Observability, and Minimality, Proceedings of the IEEE, Vol.65, No.6, June 1977, pp.945-961. S.K. Mitra, A.D. Sagar, and N.A. Pendergrass, Realizations of two-dimensional recursive digital filters, IEEE Transactions on Circuits and Systems, CAS-22, No.3, March 1975, pp.177-184. P.N. Paraskevopoulos, G.E. Antoniou, and S.J. Varoufakis, Minimal state space realization of 3-d systems, Proceedings of the IEEE, Vol. 35, No. 2, April 1988, pp.65-70. G.S. Rao, P. Karivaratharajan, and K.P. Rajappan, On realization of two- dimensional digital-filter structures, IEEE Transactions on Circuits and Systems, CAS-23, No.7, 1976, pp.479-494. M. Rosenblum, A corona theorem for countable many functions, Integral Equa- tions and Operator Theory 3, 1980, pp. 125-137. W. Rudin, Real and Complex Analysis, McGraw-Hill, New York 1966. B. Sz.-Nagy and C. Foias, Harmonic analysis of operators on a Hilbert space, North-Holland Publishing Co., Amsterdam, 1970. [25] S.J. Varoufakis, P.N. Paraskevopoulos, and G.E. Antoniou, On the minimal state-space realizations of all-pole and all-zero 2-d systems, IEEE Transactions on Circuits and Systems, CAS-34, No.3, March 1987, pp. 289-292. BIOGRAPHICAL SKETCH Andrew T. Tomerlin was born in Orlando, Florida, on July 25, 1974. He graduated from the University of Florida in August 1996 with a Bachelor of Science with Honors in physics. In December 2000 he will be graduating with his Ph.D in mathematics and a MS in electrical engineering. While at the University of Florida he met and married Paromita Bose from New Delhi, India. 45 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Scott McCullough , Chairman Associate Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. UwTKyL: ftp Murali Rao Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. M^L Li-Chien Shen Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Douglas €lianzer Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Docto/'of Philosophy. j-< adob Harjamer Professor of Electrical Engineering This dissertation was submitted to the Graduate Faculty of the Department of Mathematics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December 2000 Dean, Graduate School ID UNIVERSITY OF FLORIDA 3 1262 08555 1819